# Properties

 Label 8-882e4-1.1-c2e4-0-4 Degree $8$ Conductor $605165749776$ Sign $1$ Analytic cond. $333591.$ Root an. cond. $4.90232$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·4-s + 12·16-s − 112·23-s + 80·25-s − 112·29-s + 64·37-s − 128·43-s + 168·53-s + 32·64-s + 16·67-s − 448·71-s − 144·79-s − 448·92-s + 320·100-s − 112·107-s + 32·109-s + 56·113-s − 448·116-s − 468·121-s + 127-s + 131-s + 137-s + 139-s + 256·148-s + 149-s + 151-s + 157-s + ⋯
 L(s)  = 1 + 4-s + 3/4·16-s − 4.86·23-s + 16/5·25-s − 3.86·29-s + 1.72·37-s − 2.97·43-s + 3.16·53-s + 1/2·64-s + 0.238·67-s − 6.30·71-s − 1.82·79-s − 4.86·92-s + 16/5·100-s − 1.04·107-s + 0.293·109-s + 0.495·113-s − 3.86·116-s − 3.86·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.72·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{8} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$333591.$$ Root analytic conductor: $$4.90232$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{882} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.3510406110$$ $$L(\frac12)$$ $$\approx$$ $$0.3510406110$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$( 1 - p T^{2} )^{2}$$
3 $$1$$
7 $$1$$
good5$C_2^2:C_4$ $$1 - 16 p T^{2} + 2752 T^{4} - 16 p^{5} T^{6} + p^{8} T^{8}$$
11$C_2^2$ $$( 1 + 234 T^{2} + p^{4} T^{4} )^{2}$$
13$C_2^2:C_4$ $$1 - 320 T^{2} + 54400 T^{4} - 320 p^{4} T^{6} + p^{8} T^{8}$$
17$C_2^2:C_4$ $$1 - 688 T^{2} + 960 p^{2} T^{4} - 688 p^{4} T^{6} + p^{8} T^{8}$$
19$C_2^2:C_4$ $$1 - 900 T^{2} + 456870 T^{4} - 900 p^{4} T^{6} + p^{8} T^{8}$$
23$D_{4}$ $$( 1 + 56 T + 1770 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
29$D_{4}$ $$( 1 + 56 T + 2224 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
31$C_2^2:C_4$ $$1 - 2276 T^{2} + 2834758 T^{4} - 2276 p^{4} T^{6} + p^{8} T^{8}$$
37$D_{4}$ $$( 1 - 32 T + 2112 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$C_2^2:C_4$ $$1 - 1040 T^{2} + 3520 p^{2} T^{4} - 1040 p^{4} T^{6} + p^{8} T^{8}$$
43$D_{4}$ $$( 1 + 64 T + 3154 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
47$C_2^2:C_4$ $$1 + 1564 T^{2} + 6450886 T^{4} + 1564 p^{4} T^{6} + p^{8} T^{8}$$
53$D_{4}$ $$( 1 - 84 T + 3510 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
59$C_2^2:C_4$ $$1 - 5316 T^{2} + 17444838 T^{4} - 5316 p^{4} T^{6} + p^{8} T^{8}$$
61$C_2^2:C_4$ $$1 - 9152 T^{2} + 41434240 T^{4} - 9152 p^{4} T^{6} + p^{8} T^{8}$$
67$C_2$ $$( 1 - 4 T + p^{2} T^{2} )^{4}$$
71$D_{4}$ $$( 1 + 224 T + 22234 T^{2} + 224 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
73$C_2^2:C_4$ $$1 - 12864 T^{2} + 88688448 T^{4} - 12864 p^{4} T^{6} + p^{8} T^{8}$$
79$D_{4}$ $$( 1 + 72 T + 12210 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
83$C_2^2:C_4$ $$1 - 3876 T^{2} + 69670758 T^{4} - 3876 p^{4} T^{6} + p^{8} T^{8}$$
89$C_2^2:C_4$ $$1 - 6768 T^{2} + 136931136 T^{4} - 6768 p^{4} T^{6} + p^{8} T^{8}$$
97$C_2^2:C_4$ $$1 - 15168 T^{2} + 120056640 T^{4} - 15168 p^{4} T^{6} + p^{8} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$